Math55_fa05quizzes_bergman - Quiz 1 Problem 1 (10) Describe...

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Unformatted text preview: Quiz 1 Problem 1 (10) Describe a procedure for determining if the intersection of two finite sets ,say {a1,a2,...,an} and {171,152, Wing}, is a subset of another finite sets, say {c1,...,cm}. Express the procedure in a pseudocode. Count the number of steps it takes for the program to halt. W94“ sew? (hi-“M 7 i é»~~,i3, mes) A' i 422 W3 f2: 1:) "5) 27, 7’, "z “3 M) Z //1 C -’ 7 f,'g.>/)UJ)1 ,7 JOMW fiéggé? (€‘J/"')d"z2{er‘Jéhg/z Cu‘ ’2']: {A’J'UL'KS (1:1 {(1) “V”; [M3 a) [,3 cvn‘ébii’h'} 4355th fits ‘ [FM (I?! +9 *1 V f? a-ee A 4; ¢ 5 "LL caste/5”"? ’ | m V” J Ant-r :6 5-1“ AM 5? c, gr; an Vol...) Quiz 3 Problem 1 (5) Prove that if f is 0(9) and g is 0(h) then f is 0(h). (Write this in English sentences. Explain the meaning of each symbol that you will be using, and for each symbol explain - how or why one can get the object denoted by the symbol). (e063? Agfofé? “*7?€0CA§ ‘ K'er 296T?" Fa-nfiaxq ['j J: a : ‘Ef __ X>/L, old %jchA x>lzl_wlf1’l W’ fl: is Assam” we l 4M 4 MM 0, “A” we 7,4» 21“ MM» AP CL. / 1c. a 5. z (,2, fl sag CfA ha 1% mow filnwm. 70 ‘ C fw‘w} «[3 i 0C1) lulu”. L174 +fuk LAD/(1)441} Wlékpfifi n‘ W 1/4319 ‘ . .G i O (5/ M jAK 27177:, JAM 4 “Fox” VelmélxvlL’t’arLjrflJ/M {UM doxfi'hir. SEWLé/flp “w jaw/5 All 5:065); Twy/g/ -P )3 "4/5 )Lu m 47,“! 4'3 53k, flu gfiw’wA ml‘zm [3' $55 “372% Alm; {M tea/fi— .< (7% 90% 4' 445$) 74%; A/ WEI: LS‘; l‘Q—nv, Q’r J/i Problem 2 (5) Prove that iff : S —+ Y and if X, Z are subsets of S, then f(X U Z) : f(X) U f {Write this in Engh'sh sentences. Explain the meaning of each symbol that you will be using, and for each symbol explain how or why one can get the object denoted by the symbol). a? . SEW, flick; e (We a a me "x 4 W") Mg Y"? array fl“ or FM {LA {S ’7‘ 4M {Sara-Al}, 5? Alia] 1'? r is M WI’H‘x £511) is m ffy)‘ Inc X131“ afiga) is _/A -F(—Z—). . t o/ flnggc’lgL 5%,ch a? an; "L inc “Fa/{7‘ (Liam, 47? 5/ X) 4V0 is A chub—Fm) M m ¥(?U75). "L? X {’5 A web 3( or 27, 100K 4” X/ 940 z; 2% 01514:! ’{ “F179, arm MM é,” “PL Mei. (mum mm A 465”! 2;, regime can fly wen/1% 4H M 1"? [S [*3 1C AU?) find a" film )CCAU}):.FCX)U—PIZU ‘ J?§\ ig‘mw No [736% Quiz 4 COMMENT: This quiz is out 0f 20 points. Your grade for this quiz will be divided by 2, to, make sure that all quizzes or out of 10. Problem 1 (5) Using Euclidian Algorithm find the greatest common divisor of 385 and 924. 3‘55 cave-w u :3v5-2 % , ‘7’” ’5“ ya! (moo =77 . 15H 4?; 3?: ‘7 z + 7‘7 23%”; 1222 Problem 2 (15) Shoe that if}? is a. prime number then either 4mg 5 1(modp) has no solutions or the only solutions of it are integers 1: such that a: E 1J—jg—1(n10d;o) or x E P—Elfinodp). [Hint This is like the homework problem. First try to figure out for which p’s there are no solutions. Then show that for other p’s all solutions have to be of the given form. The last step is to show that every x of the given form is a solution. You can use the back of this paper] . ,, 40:12; \ c J"; )0 2’3 Prim “T? (fat I 5-: A; k 5a)“; V é: L Q'Jto/ l/ ‘X " (J 4“ A0? Kali-514W "‘F p fax/Lg, (ff-Ix UL? {we +L Agar: in, 112” x a, 17.417196! N P [is 0" 0J0! “MAM‘ M (Rx/5+; he 51 5 ASSQM, ah ON # a“ I41. 904%". x; A; Par». Fe; = L2 is 6% Mfg; ,( J 1 gr? .m—n- 1 OJ )4 [1 at." 1 +17 2 f; Gtzj’gfl'fl/u‘hnf. 7%. ‘ Flt,ng ’97"? “is/2"] ammfih/ Z \ / nfisém Quiz 5 Problem 1 (5) Using Fermat’s Little T hearem, Show that 368 E 13(m0d17). Pr} 474,39 t7 q 3/ fall/a 3r, :1 (Y to 1.25"“! % 99:;(MP) S n {bl/36 ,fl q, 25,;qu p-1 1:733 3 ‘3 EI’SKAHIV) 5(«(7% ’€ 7 l3 “3“”? I? Problem 2 { 5 ) Prove that there are infinitely many primes of the form 616 + 5, and explain why ' doesn’t the same proof imply that there are infinitely many primes of the form 616 + 1. atpflagy {[7 7 f), {an [A d t &I/ [army/5 a) 4 {filch'jnt ‘va [{1“; :[(&+I) —/ MM ‘ 1/ WM 29 47a». 512.73“ ff K545”. the; in 6: (Jélwfl {4m} .wKMd/V: 5 K Wait/iv" «flatten <52 amtew/ 54 74/ Vi Hwy ...
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This note was uploaded on 04/19/2008 for the course MATH 55 taught by Professor Strain during the Fall '08 term at University of California, Berkeley.

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Math55_fa05quizzes_bergman - Quiz 1 Problem 1 (10) Describe...

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